Mapping class group action on the cohomology of the ${\mathrm{S}\mathrm{L}}_n$ character variety

We describe the mapping class group action on the cohomology of the twisted ${\mathrm{S}\mathrm{L}}_n$-character variety of a surface ${\mathrm{\Sigma }}_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this allows us to reduce the problem to the mapping class group action on the cohomology of a canonical finite cover of ${\mathrm{\Sigma }}_g$, which was studied by Looijenga.